How to transfer uvs from static mesh to the same rigged and animated mesh
Select the object you want to copy UV map to, then the source object. Execute Ctrl-L > Transfer UV Layouts. If you're using an older version of Blender, it's in UV Maps property group's dropdown button. This will work flawlessly only if both are exactly the same, no vertices/edges added or removed.



You still need to edit the destination object's texture to use the right UV map, afterwards.

Edit: In your case, it seems like both character meshes are attached to different image files. Two identical objects can have identical UV maps, material and texture, but look different in the viewport because they're attached to different image files:



To fix this, make sure the image file being displayed on Image Editor is the same, when both objects are edited. This doesn't affect how they're rendered, though.